Problem 2 - Algebra 2 and Trigonometry

Question

Explain why the solution set of \(2 \csc ^{2} \theta-\csc \theta=0\) is the empty set.

Step-by-Step Solution

Answer

The solution set is empty because \(\csc \theta = \frac{1}{2}\) implies an impossible condition for \(\sin \theta\) and \(\csc \theta = 0\) does not exist.

Step 1: Understand the Equation

The given equation is \(2 \csc^2 \theta - \csc \theta = 0\). Recall that \(\csc \theta = \frac{1}{\sin \theta}\), so our task is to find values of \(\theta\) for which this equation holds.

Step 2: Substitute and Simplify

Substitute \(\csc \theta = x\), which gives us the equation \(2x^2 - x = 0\). This transformation helps to focus on finding solutions related to \(\csc \theta\) rather than \(\theta\) directly.

Step 3: Factor the Quadratic Equation

Factor the equation \(2x^2 - x = 0\) to find \(x(2x - 1) = 0\). The solutions for \(x\) are \(x = 0\) and \(2x - 1 = 0\).

Step 4: Solve for x

For the second equation \(2x - 1 = 0\), solving gives \(x = \frac{1}{2}\). Thus, we have potential solutions \(x = 0\) and \(x = \frac{1}{2}\).

Step 5: Analyze the Solutions in Terms of \(\csc \theta\)

\(\csc \theta = 0\) does not exist because \(\csc \theta = \frac{1}{\sin \theta}\), and sine cannot be zero in the denominator. As for \(\csc \theta = \frac{1}{2}\), this implies \(\sin \theta = 2\), which is impossible since the range of sine is \([-1, 1]\).

Step 6: Conclude the Solution Set

Since neither \(\csc \theta = 0\) nor \(\sin \theta = 2\) provide valid trigonometric solutions, the solution set is empty.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is the reciprocal of the sine function. This means that \( \csc \theta = \frac{1}{\sin \theta} \). Understanding this relationship is crucial when dealing with trigonometric equations, as it allows for transforming and solving equations in terms of more familiar functions like sine.
The cosecant function is undefined for angles where \( \sin \theta = 0 \), which occur at integer multiples of \( \pi \) (e.g., 0, \( \pi \), 2\( \pi \), etc.). At these points,

• The sine value becomes zero, leading to division by zero in the cosecant function.
• Consequently, \( \csc \theta \) approaches infinity or "is undefined" at these angles.

This behavior makes understanding the constraints of \( \csc \theta \) critical when finding solutions for trigonometric equations involving this function.

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. In the context of the exercise, after substituting \( \csc \theta = x \), the trigonometric equation transforms into a quadratic equation: \( 2x^2 - x = 0 \).
This transformation simplifies the challenge of solving equations for trigonometric functions by allowing us to use algebraic methods. Specifically, the "zero-product property" is applied here.
To solve \( 2x^2 - x = 0 \):

• Factor out the common factor, \( x \), resulting in \( x(2x - 1) = 0 \).
• This gives potential solutions for \( x \): \( x = 0 \) or \( 2x - 1 = 0 \).

However, not all algebraic solutions correspond to valid trigonometric solutions, as in our exercise where neither \( \csc \theta = 0 \) nor \( \csc \theta = \frac{1}{2} \) yields values within the sine function's possible range.

Trigonometric Identities

Trigonometric identities are mathematical relationships between the angles and sides of a triangle in trigonometry. They serve as fundamental tools for simplifying expressions and solving equations.
Notably, the reciprocal identities express each of the trigonometric functions as a reciprocal of another. In this context:

• The cosecant function is given as \( \csc \theta = \frac{1}{\sin \theta} \).
• Recognizing that \( \csc \theta \rightarrow x \) simplifies manipulation of the equation by temporarily transforming the trigonometric problem into an algebraic one.

Thus, trigonometric identities help bridge the gap between algebraic solutions and their geometric meaning. They highlight why certain outcomes, such as \( \sin \theta = 2 \), are not feasible. They remind us that sine's range between \(-1\) and \(1\) imposes limits on what \( \csc \theta \) and other identities can achieve. These concepts underscore the absence of a solution within the provided equation's context.